So just to follow up on this for future reference,
Airberlin/Niki will not reimburse my hotel booking, even though they changed my airport of departure. They say that they can't give me my money back for booking the hotel.
I'm not impressed, and will remember this in the future when thinking of booking with Airberlin. This is not very professional of them.
They have agreed to rebook me on a later flight, which gives me time to travel from MXP to LIN by bus/train. Unfortunately, I have been booked on an Alitalia flight, which means that I won't have access to the oneworld lounge.
This is my first bad experience with Airberlin/Niki, but it seems like they think they can change flights without compensating customers for inconvenience. Not a good result.